相关论文: Normal form of D4-plus singularities of fronts and…
We give useful and simple criteria for determining D_4 singularities of wave fronts. As an application, we investigate behaviors of singular curvatures of cuspidal edges near D_4^+ singularities.
We construct a form of swallowtail singularity in R^3 which uses coordinate transformation on the source and isometry on the target. As an application, we classify configurations of asymptotic curves and characteristic curves near…
To study a deformation of a singularity taking into consideration their differential geometric properties, a form representing the deformation using only diffeomorphisms on the source space and isometries of the target space plays a crucial…
We give a formula to count the number of $D_4$ singularities in a stable frontal perturbation of a corank $2$ wave front singularity $f\colon (\mathbb{C}^3,0) \to (\mathbb{C}^4,0)$ using Mond's method of stable perturbations of map germs.…
We study singularities of Gauss maps of fronts and give characterizations of types of singularities of Gauss maps by geometric properties of fronts which are related to behavior of bounded principal curvatures. Moreover, we investigate…
We investigate necessary and sufficient conditions for the extendibility and boundedness of Gaussian curvature, Mean curvature and principal curvatures near all types of singularities on fronts. We also study the convergence to infinite…
Let X be a smooth complex variety and Y be a closed subvariety of X, or more generally, a closed subscheme of X. We are interested in invariants attached to the singularities of the pair (X, Y). We discuss various methods to construct such…
We give criteria for which a principal curvature becomes a bounded $C^\infty$-function at non-degenerate singular points of wave fronts by using geometric invariants. As applications, we study singularities of parallel surfaces and extended…
This is a survey article on recognition problem of frontal singularities. We specify geometrically several frontal singularities and then we solve the recognition problem of such singularities, giving explicit normal forms. We combine the…
We define a frontal bundle by imposing a compatibility condition on two types of coherent tangent bundles over a surface with boundary. Since it is known that there are two Gauss-Bonnet type formulas for coherent tangent bundles, we obtain…
We shall introduce the singular curvature function on cuspidal edges of surfaces, which is related to the Gauss-Bonnet formula and which characterizes the shape of cuspidal edges. Moreover, it is closely related to the behavior of the…
An intriguing correspondence between four-qubit systems and simple singularity of type $D_4$ is established. We first consider an algebraic variety $X$ of separable states within the projective Hilbert space…
We study the flat geometry of the least degenerate singularity of a singular surface in $\mathbb R^4$, the $I_{1}$ singularity parametrised by $(x,y)\mapsto(x,xy,y^{2},y^{3})$. This singularity appears generically when projecting a regular…
In this survey article we introduce the notion of frontals, which provides a class of generalised submanifolds with singularities but with well-defined tangent spaces. We present a review of basic theory and known studies on frontals in…
We give a normal form of the cuspidal edge which uses only diffeomorphisms on the source and isometries on the target. Using this normal form, we study differential geometric invariants of cuspidal edges which determine them up to order…
In this paper, we prove a similar result to the fundamental theorem of regular surfaces in classical differential geometry, which extends the classical theorem to the entire class of singular surfaces in Euclidean 3-space known as frontals.…
In this paper, we discuss the recognition problem for A_k-type singularities on wave fronts. We give computable and simple criteria of these singularities, which will play a fundamental role in generalizing the authors' previous work "the…
We investigate singularities of all parallel surfaces to a given regular surface. In generic context, the types of singularities of parallel surfaces are cuspidal edge, swallowtail, cuspidal lips, cuspidal beaks, cuspidal butterfly and…
We characterize the extendibility of the normal curvature on frontals and we give a representation formula of this type of frontals. Also we give representation formulas for wavefronts on all types of singularities and others sub classes of…
Recently, a novel four-dimensional Gauss-Bonnet theory has been suggested as a limiting case of the original $D$-dimensional theory with singular Gauss-Bonnet coupling constant $\alpha\rightarrow\alpha/(D-4)$. The theory is proposed at the…